Hi, I’ve been stuck on this problem for a week and can’t seem to find any example that would help me to solve it. I would really appreciate it if someone would explain the steps to solving this problem. Thank you so much.
Hi, I’ve been stuck on this problem for a week and can’t seem to find any example that would help me to solve it. I would really appreciate it if someone would explain the steps to solving this problem. Thank you so much.
I developed a theorem (imcomplete) sometime ago, that if you have an integral linear combination of irrational distinct radicals equal to each other then there corresponding parts are equal.
For example, if
where are integers then,
.
Now if you have,
where are integers is just impossible,
If you have that,
where are integers then,
Thus,
Then,
is just impossible in integers.
This is my reasoning that why I would not expect a simpler solution with the integers.
No I did not make a mistake in what I said.Originally Posted by CaptainBlack
Note I made a mistake in doing
the wrong problem.
If you used my method then you will see that it works
-------------------------
I base it on the fact that,
is always irrational whenver all are integers and the b_i are all non-squares and a_i are not zero.
Then given
=
You can write,
where
Which is rational an impossibility unless a_i=c_i giving zeros.
You can use my method which I shown, (I only did the wrong example which is why it did not work).Originally Posted by Ranger SVO
You attempt to find integers such as,
Thus,
Thus,
Then, by my previous post,
Look at the second equation,
This, is equivalent to
Since are integers the only possibilities are,
Now we check which one of these satisfies the second equation,
we see that,
Thus,
Both of these are true, indeed,
But we need to chose,
Because square roots are non-negative and,